POLYPHASE CIRCUITS (Delta, Wye, Power, Power Factor Correction) (Pictures Only)

Sample Problems on Youtube:

IRWIN 11.10: A three phase very simple exercise.

IRWIN 11.14: A three phase exercise, delta load

IRWIN 11.27: Three Phase Tutorial, currents in a delta.

Wyte to Delta Conversion

Y to  Conversions
In terms of power, currents & line voltages, the following sources are the same and may be used interchangably in most cases. Note, the Y connection should be used in a one-line diagram.  Wye connected source           
Figure 8: A Y Source
VA = Vab / (√3 <30°)
to
Delta connected source
Figure 9: A Delta Source
Vab = VA √3(<30°)

Similarly, the two loads given below are the same in terms of the resulting power, line currents and line voltages and can usually be substituted as desired. Note that the Y connection is the one needed for the one-line diagram!

Wye connected load

Figure 10: A Y Impedance Load to

Delta connected load

Figure 11: A  Impedance Load

Three Phase Circuit Wye and Delta Basics

3 Phase WYE (Y)

A 3 Phase-Wye connected system consists of three hot lines, or phases, commonly referred to as X, Y, Z, a neutral, and a ground wire for a total of five wires in a power distribution cable.

In North America the most common 3 Phase-Y voltages are either 120/208 VAC or 277/480 VAC, while internationally the most common 3 Phase voltage is 230/400 VAC. The lower voltage in each case is the country’s standard utilization voltage and is measured Line-to-Neutral, while the higher voltage is measured Line-to-Line. The Line-to-Line voltage is always 1.732 times higher than the Line-to-Neutral voltage in a Wye configured 3 Phase system.

The line current supplied to the load is also the same as the phase current. It is important to note that when all three “Hot” phases of the system are loaded equally, the net current draw in the neutral line is zero!

3 Phase Delta (Δ)

A 3 Phase-Delta connected system consists of three hot lines, commonly referred to as X, Y, Z, and a ground wire for a total of four wires in a power distribution cable.

In North America the most common 3 Phase-Δ voltages are either 208VAC or 240VAC, while internationally the most common 3 Phase voltage is 230 VAC. These phase voltages are measured Line-to-Line and are typically the country’s standard utilization voltage.

Since there is no neutral line in a Delta-connected system, there is no Line-to-Neutral voltage! However, the line current in a Delta-connected system is 1.732 times the phase current supplied to the load(s).

Proper care must be taken to correctly size cables in a Delta system because the line currents are much higher than the load (or phase) currents.

Delta systems typically have lower line voltages but higher line currents than Wye-connected systems.

Apparent Power and Power Factor

True, Reactive, and Apparent power

We know that reactive loads such as inductors and capacitors dissipate zero power, yet the fact that they drop voltage and draw current gives the deceptive impression that they actually do dissipate power. This “phantom power” is called reactive power, and it is measured in a unit called Volt-Amps-Reactive (VAR), rather than watts. The mathematical symbol for reactive power is (unfortunately) the capital letter Q. The actual amount of power being used, or dissipated, in a circuit is called true power, and it is measured in watts (symbolized by the capital letter P, as always). The combination of reactive power and true power is called apparent power, and it is the product of a circuit's voltage and current, without reference to phase angle. Apparent power is measured in the unit of Volt-Amps (VA) and is symbolized by the capital letter S.

As a rule, true power is a function of a circuit's dissipative elements, usually resistances (R). Reactive power is a function of a circuit's reactance (X). Apparent power is a function of a circuit's total impedance (Z). Since we're dealing with scalar quantities for power calculation, any complex starting quantities such as voltage, current, and impedance must be represented by their polar magnitudes, not by real or imaginary rectangular components. For instance, if I'm calculating true power from current and resistance, I must use the polar magnitude for current, and not merely the “real” or “imaginary” portion of the current. If I'm calculating apparent power from voltage and impedance, both of these formerly complex quantities must be reduced to their polar magnitudes for the scalar arithmetic.

There are several power equations relating the three types of power to resistance, reactance, and impedance (all using scalar quantities):

Resistive load only:

True power, reactive power, and apparent power for a purely resistive load.

Reactive load only:

True power, reactive power, and apparent power for a purely reactive load.

Resistive/reactive load:

True power, reactive power, and apparent power for a resistive/reactive load.

These three types of power -- true, reactive, and apparent -- relate to one another in trigonometric form.

Power triangle relating appearant power to true power and reactive power.

Using the laws of trigonometry, we can solve for the length of any side (amount of any type of power), given the lengths of the other two sides, or the length of one side and an angle.

Effective or RMS Value

• We have a few different ways to specify the size of an ac current or voltage.
• We can give either
  • the peak value, or
  • the peak-to-peak value, or
  • something called the effective value (also called rms value).
• Example: As you can see, the sine wave shown below has a peak voltage of 6 V_p. Also, its peak-to-peak voltage is 12 V_pp. And, as we'll see below, it's effective voltage is 4.24 V_rms. So you can't just say something like "The sine wave had a voltage of 6 V." You've got to be careful to say whether you're talking about peak voltage, peak-to-peak voltage, or effective voltage.
   
• This may seem confusing, but you have to be able to deal with it. It's similar to the situation that we have with temperatures or distances: when you give the distance between two cities, you can give it either in miles or in kilometers. It's the same distance, but expressed with two different units.
• These distinctions apply only to ac, not to dc.

RMS Value (Effective Value)

• The root-mean-square (rms) value or effective value of an ac waveform is a measure of how effective the waveform is in producing heat in a resistance.
• Example: If you connect a 5 V_rms source across a resistor, it will produce the same amount of heat as you would get if you connected a 5 V dc source across that same resistor. On the other hand, if you connect a 5 V peak source or a 5 V peak-to-peak source across that resistor, it willnot produce the same amount of heat as a 5 V dc source.
• That's why rms (or effective) values are useful: they give us a way to compare ac voltages to dc voltages.
• To show that a voltage or current is an rms value, we write rms after the unit: for example, V_rms = 25 V rms.

Maximum Average Power Transfer

The maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.

RL= Re{ZTh} and XL = - Im{ZTh}

The maximum power in this case:

Pmax =  

Where V2Th and I2N represent the square of the sinusoidal peak values.

We’ll next illustrate the theorem with some examples.

Example 1

R1 = 5 kohm, L = 2 H, vS(t) = 100V cos wt, w = 1 krad/s.

a) Find C and R2 so that the average power of the R2-C two-pole will be maximum

b) Find the maximum average power and the reactive power in this case.

c) Find v(t) in this case.

The solution by the theorem using V, mA, mW, kohm, mS, krad/s, ms, H, m F units:v

a.) The network is already in Thévenin form, so we can use the conjugate form and determine the real and imaginary components of Z_Th:

R₂ = R₁ = 5 kohm; wL = 1/w C = 2 ® C = 1/w²L = 0.5 mF = 500 nF. 

b.) The average power: 

P_max = V²/(4*R₁) = 100²/(2*4*5) = 250 mW

 The reactive power: first the current: 

I = V / (R₁ + R₂ + j(wL – 1/wC)) = 100/10 = 10 mA

Q = - I²/2 * X_C = - 50*2 = - 100 mvar

c.) The load voltage in the case of maximum power transfer: 

V_L = I*(R₂ + 1/ (j w C ) = 10*(5-j/(1*0.5)) =50 – j 20 = 53.852 e ^{-j 21.8°} V

and the time function: v(t) = 53.853 cos (wt – 21.8°) V

Instataneous Power and Average Power

INSTANTANEOUS POWER AND AVERAGE POWER

Power is the most important quantity in electric utilities, electronics . . . because such system involve transmission of power from one point to another.

Instantaneous power is the power at any instant of time. In Ac circuit the instantaneous electric power is given by 

P= VI

but these quantities are continuously varying. Instantaneous Power p(t) is the power, p(t)= u(t)*i(t). It is the product of the time functions of the voltage and current. This definition of instantaneous  power is valid for signals of any waveform. The unit for instantaneous power is VA.

REAL and ACTIVE POWER

P can be define in two ways: as the real part of the complex power or as the simple average of the instantaneous power. The second definition is more general because with it we can define the instantaneous power for any signal waveform, not just sinusoids.

The instantaneous power at any time t can be expressed as:

Pinstantaneous= VmImsinωtsin(ωt-ɸ)

Average power is more convenient to measure. It is the average of the instantaneous power over one period. And it is given by:

Pave= VIcosɸ

where ɸ is the phase angle between the current and the voltage and where V and I are understood to be the effective or rms of the voltage and current. The term cos ɸ is called the power factor for the circuit.